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    Chromatic Number

    Basic definition

    The chromatic number is the smallest number of colors necessary to color the vertices of a graph or the regions of a surface such that no two adjacent vertices or regions are the same color.

    Detailed definition

    The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color, i.e., the smallest value of k possible to obtain a k-coloring. Minimal colorings and chromatic numbers for a sample of graphs are illustrated above. The chromatic number of a graph G is most commonly denoted χ(G) (e.g., Skiena 1990, West 2000, Godsil and Royle 2001, Pemmaraju and Skiena 2003), but occasionally also γ(G). Empty graphs have chromatic number 1, while non-empty bipartite graphs have chromatic number 2.

    Related terms

    Betti number | bicolorable graph | Brelaz's heuristic algorithm | Brooks' theorem | chromatic invariant | chromatic polynomial | cyclic chromatic number | edge chromatic number | edge coloring | Euler characteristic | fractional chromatic number | genus | Heawood conjecture | k-chromatic graph | k-colorable graph | map coloring | perfect graph | three-colorable graph | torus coloring

    Educational grade level

    college level

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